Skyscrapers sway in high-wind conditions. in one case, at t=2s, the top floor swayed 30cm to the left (-30cm), and at t=12, the top floor swayed 30 cm to the right (+30cm) of its starting position.

Question

a) what is the equation of a sinusoidal function that describes the motion of the building in terms of time?

b) if a damper is added to this building, it will reduce the sway (not the period) by 70%. what is the equation of the new function that describes the motion of the building in terms of time?

Answer 1

I assume that the building goes directly from one extreme to the other in 12-2=10 seconds. Therefore the half-period is 10 seconds, or the period T=20 s.

The neutral position is at (2+10)/2=6 seconds, when the displacement is zero.

Thus we have
Xo=30 cm = 0.3m
X(t)=Xo sin((t-6)*2π/T), or
X(t)=0.3sin((t-6)π/10)

Answer 2

a) |a|= 30 cm
1/2 period = 10
period = 20
k= 360/ period
= 360/20
= 18
p = 2
a = -30 (as a reflected cosine)
therefore, d(t)= -30 cos{18(t-2)}
b) Reduce sway = reduce of amplitude
|a|= 100% - 70%
= 30%
|a| = 30 (.3)
= 9 cos{ 18(t-2)}